Recent development has enabled fibre and matrix failure in a fibre reinforced composite material to be predicted separately. Matrix yield/failure prediction is based on a Von Mises strain and first strain invariant criteria. Alternative matrix failure criteria for enhanced prediction accuracy are discussed in this paper. The proposed failure envelope formed with basic failure criteria intersects with uniaxial compression, pure shear and uniaxial tensile test data points smoothly. For failure of typical neat resin, significant improvement of prediction accuracy compared with measured material data is demonstrated. For a unit cell with a fibre and surrounding matrix with typical material properties, a FEM analysis indicates a significant improvement in prediction accuracy in the pure shear load case and a marginal improvement in the biaxial tensile load case. This paper also provided a preliminary discussion about the issues when material nonlinearity of the matrix material is involved.
Fibre reinforced polymer matrix composites are being increasingly used for aircraft structures because of their superior structural performance (such as high strength, high stiffness, long fatigue life, and low density). Some recently developed military helicopters have nearly all-composite airframe structures (such as Eurocopter Tiger and Bell/Boeing V22).
In the conventional laminate theory widely used to predict the strength of fibre reinforced polymer composites, the laminae are treated as homogeneous orthotropic materials. Recent development has made it possible to extend the conventional laminate theory to predict separately the failure of the polymer matrix and fibres. In essence, this is through a microstructural analysis conducted on a unit cell of the composite material that contains a fibre and surrounding polymeric matrix to determine the correlation between the stress-strain states of the whole cell and its matrix and fibre components. This correlation is then used in a structural analysis to predict matrix or fibre failure. In a linear finite element method, the failure of the polymer matrix and fibres can be separately predicted during postprocessing of the results from a computation based on the conventional laminate theory, by correlating the element of stress-strain state with matrix and fibre stress-strain states. In principle, this micromechanical approach may also predict the interfacial failure between the fibre and matrix.
Significant work in the aforementioned area was reported by Gosse and his coworkers [
These two strain variants are considered to be able to indicate matrix initial failure due to volume increase (dilational strain) and distortional strains, respectively. When either of these reaches its critical value, failure will occur. Typical matrix initial failure includes microcracking and delamination initiation. Note that (
As a newly developed novel approach alternative to the conventional laminate theory, this method has great potential to be further developed, both in the areas of verification and validation. The analytical approach can be further assessed and improved. A wide range of tests need be conducted to validate the model or determine its limitations.
This paper aims to demonstrate that the accuracy of matrix failure prediction could be increased significantly by enhancing these two failure criteria. In addition, this paper will also discuss the issues when material nonlinearity of the matrix material is involved. Note that the approach provided in [
In this section, the discussion is restricted to the linear elastic condition (the discussion about material nonlinearity is provided in Section Uniaxial compression
Pure in-plane shear
Uniaxial tension
Biaxial tension
where
Four basic load cases considered.
According to [
It is well known that the strength of polymer materials could be affected by compressive hydrostatic stress [
Table
Typical resin material strength data [
Resin | Tensile (MPa) | Compression (MPa) | Shear (MPa) | Compression (%) | Shear (%) | ||
---|---|---|---|---|---|---|---|
Type 1 |
Yield strength | 58 | 96 | 50 | |||
Ultimate strength | 58 | 130 | 62 | Ultimate strain | 4.5 | 9.3 | |
| |||||||
Type 2 |
Yield Strength | 50 | 100 | 36 | |||
Ultimate strength | 50 | 130 | 60 | Ultimate strain | 5.9 | 7.0 |
For the prediction of yield strength, if one generates the critical value of
Failure is predicted when
Equations (
For isotropic materials, a stress based failure criterion is also commonly used since it is often easier to use and has a clearer physical meaning (e.g., hydrostatic pressure effect shown in Drucker-Prager criterion). Alternative equations of (
Parameter
Resin | Strain-based equation | Stress-based equation | ||
---|---|---|---|---|
|
|
|
| |
Type 1 |
|
0.439 | 86.6 MPa | 0.098 |
Type 2 |
|
1.69 | 62.4 MPa | 0.376 |
Before discussing the uniaxial and biaxial tension cases, we may consider an extreme case where the polymer is loaded with uniformly distributed tensile stress in all the three axial directions. It is well known that the Von Mises yield criterion is not valid in this situation. A Drucker-Prager criterion established using parameters determined from compression and shear load cases would also significantly overpredict the strength in this dilation type failure situation.
The failure pattern of typical composite matrix materials under uniaxial loading can be considered as dilation failure (rather than shear or distortional failure), as indicated from the observation that the broken sections generally are approximately perpendicular to the tensile load direction (rather than with a lager angle, i.e., shear or distortion angle), yet without significant necking.
Equation (
Failure envelope for a polymer described in [
From Figure
Bardenheier [
Data from neat resin biaxial tests [
Resin | Stress applied* | Error with prediction method** (%) | |||
---|---|---|---|---|---|
|
|
Equation ( |
Equation ( |
Max stress criterion | |
Type A | 0.79 | 0.79 | −36.7 | −10.5 | +21.0 |
Type B | 0.84 | 0.50 | −25.4 | +2.3 | +16.0 |
Type C | 0.85 | 0.80 | −39.4 | −14.3 | +15.0 |
*Expressed using the ratio of the yield stress in the biaxial test against that in the uni-axial tensile test.
**In each method, the yield stress from the uni-axial tensile test was used to determine the critical value for prediction of yield strength in biaxial tests. A “+” sign means overprediction and a “−” sign means underprediction.
A generic form of the equation that expresses the failure envelop can be as follows:
Equations (
As mentioned earlier, though there are many advanced material models available, only basic failure criteria that are relatively easy to implement are considered here. Figure
Illustration of failure envelope (plane-stress). UT = uniaxial tensile; BT = biaxial tensile; PS = pure shear; UC = uniaxial compression;
This figure shows that the portion of the failure envelope formed with the Drucker-Prager failure criterion can intersect with uniaxial compression, pure shear and uniaxial tensile test data points and join with the
In the 3-dimensional space with
A unit cell analysis using a finite element method was further conducted with the four typical load cases discussed previously. The failure prediction based on (
Since the discussion in Section
The following typical material properties are assumed: fibre Young’s modulus ( fibre Poisson ratio ( matrix Young’s modulus ( matrix Poisson ratio ( Fibre volume fraction (
Referring to Figure
Unit cells with two orientations (
Orientation 1
Orientation 2
A unit cell FEM mesh (one-eighth model of the unit cell in Figure
The load condition is applied in the form of uniform surface displacement. Symmetry boundary conditions are applied to three orthogonal surfaces. The remaining unloaded surfaces were given a uniform displacement that results in the gross load on the surface being zero, to reflect the “Poisson effect.” The loading and boundary conditions applied ensure all the surfaces are kept straight, as required for the simulation of a unit cell in a lamina structure. This method was benchmarked against the rule of mixtures and proved to be accurate. Further discussion about this could be found in [
The unit cell shown in Figure
Stress concentration areas in uniaxial compression and “pure shear” load cases.
In contrast, in the pure shear case, high compressive and tensile stress concentration occurred separately at the areas shown in Figure
With a load applied that resulted in 0.01 gross strain, the calculated maximum effective stress values are listed in Table
Predicted maximum
(1) Uni-axial compression | (2) Pure shear | Ratio (2)/(1) | Discrepancy | |
---|---|---|---|---|
|
0.0574 | 0.0624 | 1.09 | 60.6% |
|
|
|
1.75 |
For practical applications of the approach described in [
Figure
Predicted maximum
(1) Uni-axial tension | (2) Biaxial tension | Ratio (2)/(1) | Discrepancy | |
---|---|---|---|---|
|
0.0376 | 0.0438 | 1.18 | 3.4% |
|
|
|
1.14 |
Stress concentration areas in uniaxial and biaxial tension load cases—orientation 1.
The unit cell shown in Figure
Predicted
(1) Uni-axial tension | (2) Biaxial tension | Ratio (2)/(1) | Difference | ||
---|---|---|---|---|---|
Point |
|
0.0141 | 0.0334 | 2.37 | 1.0% |
|
|
|
2.32 | ||
| |||||
Point |
|
0.00978 | 0.0435 | 4.44 | 49.0% |
|
|
|
2.98 | ||
| |||||
Max |
|
0.0141 | 0.0435 | 3.09 | 6.1% |
|
|
|
2.90 |
Stress concentration areas in uniaxial and biaxial tension load cases—orientation 2.
As shown in Table
Note that when a further tensile load is applied along the fibre direction, resulting in a more uniformed stress state among the three principal stresses at the critical location, a higher difference was predicted between these two criteria. Since this is beyond the four load cases considered, detailed description is not presented.
Once the matrix stress-strain state is within the nonlinear range, its modulus would no longer be constant [
Nonlinear stress-strain relationship.
In principle, a nonlinear structural analysis can be conducted using a finite element method with laminar nonlinear modulus properties based on the conventional laminate theory. The failure of the polymer matrix and fibres can be separately predicted by correlating the element stress-strain state with matrix and fibre stress-strain states at each iteration step. However, this could become impractical due to the computational expense of the nonlinear microstructural analysis at every iteration step.
Where the effect of material nonlinearity on laminate strength prediction is most apparent would be a strong matrix dominant case, such as to predict delamination on-set load for a laminate subject to a through-thickness shear load. The material nonlinearity is reflected directly in the load-displacement relationship of the laminate and thus needs to be taken into account for an accurate microstructure analysis and accurate prediction for the load limit of the laminate.
It would be feasible to conduct a global analysis run with the conventional laminate theory model (may include material nonlinearity) and give an initial assessment to matrix failure based on the “standard” linear microstructural analysis approach to locate the critical area and the local load condition. Then conduct a local model FEM structural analysis in conjunction with a unit cell microstructural analysis, in which the nonlinear material property is considered. The computational expense might be manageable.
We may examine a typical fibre dominant load case, a panel made of AS4/3501-6 prepreg tape with a quasi-isotropic layup [0/-45/45/90]s, under a uniaxial compression in-plane load along the 0-direction.
The material properties are listed in Table
Mechanical properties of AS4/3501-6 lamina—source [
Elastic constants | Strength properties | ||
---|---|---|---|
Longitudinal modulus, |
142 | Longitudinal tensile, |
2280 |
Transverse modulus, |
10.3 | Longitudinal compressive, |
−1769* |
Shear modulus, |
7.2 | Transverse tensile, |
57 |
Poisson’s ratio, |
0.27 | Transverse compressive, |
−228 |
Shear, |
71 |
*Value adjusted according to microbuckling consideration [
Shear stress-strain curve of AS4/3501-6 (±45° tension test) [
An alternative way of linear approximation is to consider that the laminate failure occurs when the 0-ply ultimate stress is reached, when the angle ply has not yet reached its ultimate strain (according to Figure
Though arguably 5% overprediction may not be negligible, the aforementioned does indicate that, with a linear approximation, when the ultimate strain of a matrix dominant lamina stress-strain component is used, the predicted loading capacity is much closer to the nonlinear analysis than that when the ultimate stress is used (35% under prediction). This is due to the matrix strain hardening behaviour and the insignificant contribution of the matrix dominated stress component in the overall loading capacity of the laminar.
The previous discussion relates two different ways of linear approximation. Extending this discussion, one may conclude that in the microstructural analysis (postprocession of a FEM linear approximation analysis) to use the ultimate strain of a matrix would result in a nonconservative but much smaller strength prediction error than to use the ultimate stress of the matrix (conservative, much larger strength prediction error).
Thus, when a prediction based on linear analysis approximation is used in the case where the matrix material nonlinearity effect is significant, it is important to determine the maximum possible error and if it is acceptable in terms of the requirement/purpose of the prediction.
A further point to make is the different way to make the linearization. As shown in Figure
Different ways of linear approximation for nonlinear stress-strain relationship of polymer matrix or matrix dominant lamina property.
Recent development has enabled fibre and matrix failure in a fibre reinforced composite material to be predicted separately. Matrix yield/failure prediction is based on a Von Mises strain and first strain invariant criteria. Improvement of the matrix failure criteria for enhanced prediction accuracy is discussed and demonstrated in this paper.
For two typical resin materials considered, Von Mises yield criterion is unable to fit both yield strength values from uniaxial compression and pure shear tests. When calibrated using the measured shear yield strength, the discrepancy between the predicted and measured uniaxial compression strengths is found to be 11% and 60%, respectively, whilst using a Drucker-Prager criterion, these discrepancies could be removed.
Use of the critical value of the first invariant strain, when calibrated using the uniaxial tensile yield strength, underpredicts the biaxial tensile strength significantly by over 30% on average, compared with available measurement data of neat resin materials. A revised criterion proposed in this paper could reduce the discrepancy to less than 10%. The proposed failure envelope intersects with all the uniaxial compression, pure shear and uniaxial tensile test data points, and a revised tensile failure criterion. The areas governed by these criteria join each other smoothly.
For a unit cell with a fibre and surrounding matrix from a lamina with 50% fibre volume fraction and typical material properties, a FEM analysis conducted in this study indicates that the difference between the yield strength of the matrix material predicted using Von Mises and Drucker-Prager type criteria is over 60% in the pure shear load case, when the critical values of these yield criteria are determined in the uniaxial compression load case.
The FEM analysis showed that the difference between the yield strength of the matrix material in a unit cell predicted using the first strain variant and the revised criterion reaches 6.1% in the biaxial tensile load case, when the critical values of these yield criteria are determined in the uniaxial tensile load case. With a tensile load added along the fibre direction, this difference is further increased.
This paper also provided a preliminary discussion about the issues when matrix material nonlinearity is involved.
As a newly developed novel approach alternative to the conventional laminate theory, this method has great potential to be further developed, both in the areas of verification and validation. The analytical approach can be further assessed and improved. A wide range of tests need be conducted to validated the model or determine its limitations.
The authors would like to acknowledge that this paper is the outcome of a research project of the Cooperative Research Centre for Advanced Composite Structures (CRC-ACS).